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Introduction To Probability 1st Edition Mark Daniel Ward, Ellen Gundlach - Solutions

Consider a pile that has 9 coins altogether. Exactly 3 of the coins are dimes (worth 10 cents each), and the other 6 coins are pennies (worth 1 cent each). Emily picks up 4 of the coins blindly (all possibilities are equally likely) without replacement. Find the expected value (in cents) of the
As in Exercises 10.2 and 11.2, four students order noodles at a certain local restaurant. Their orders are placed independently. Each student is known to prefer Japanese pan noodles 40% of the time. Let X be the number of students who order Japanese pan noodles. What is the variance of X?
Roll two dice; let X denote the maximum of the two values that appear.a. Find E(X).b. Find E(X2).c. Find Vax(X).
Three hundred little plastic yellow ducks are dumped in a pond; one of them contains a prize stamped on the bottom. Leonardo examines each duck until he discovers the prize. He discards each duck without a prize after he checks it, so that he never needs to check a duck more than one time. Find the
Let X be the number of is that appear.a. Find E(X).b. Find E(X2).c. Find Var(X).
Suppose that the mass of X is PX (1) = 0.12 PX (3) = 0.37 PX (27) = 0.42 PX (31) = 0.09 a. Find E(X). b. Find E(X2). c. Find Vary(X). d. Find E(X2 - 4X).
Let X be the number of smoke alarms in an apartment. From government data, you find the mass of X is px(0) = 0.05 px(l) = 0.15 px(2) = 0.35 px(3) = 0.30 Px(A) = 0.10 Px(5) = 0.05 Assume each alarm requires one battery per year, and batteries cost $2.35. a. What is the expected cost in batteries for
Michael plays a random song on his iPod. He has 2,781 songs, but only one favorite song. Let X be the number of songs he has to play on shuffle (songs can be played more than once) in order to hear his favorite song.a. Find E(X).b. Find E(X2).c. Find Var(X).
There are 5 horses running in a race, and you own the one named Rosie. Let X be the order Rosie finishes the race (1st, 2nd, etc.). Below is a table for the probability for each x-valuea. Graph the probability mass function.b. What place is Rosie expected to finish?c. What is the standard deviation
Which of the following statements are true for discrete random variables? Explain why each statement is true or false. a. The mean can be negative. b. The mean can be zero. c. The standard deviation can be negative. d. The standard deviation can be zero. e. The standard deviation can be greater
Suppose an acquaintance has a trick coin that comes up a head 75% of the time. He wants to play a game where he tosses the coin 3 times, and for each time the coin comes up a head, he will pay you $2. He wants to charge you a fee to play the game. If you are willing to play the game only if the
Chet the dog likes to dig holes in the yard. He has buried bones in some of the holes, but others he dug just for fun. He covers them back up with a little mound of dirt when he is done digging. The yard has 20 holes that have been filled up, and 8 of them contain bones. Chet decides to randomly
Using the cumulative distribution function for the discrete random variable X given below:a. What is the probability that X = 45?b. What is the probability mass function for X?c. Is your answer to part b a valid probability mass function? Justify your answer.d. What is the probability that X is
Realtor sales, part 1. Daily sales records for a realtor's office show that the realtor will sell 0, 1, or 2 homes in a week with the probabilities listed below:a. Is this a valid probability mass function? Why or why not?b. What is the expected number of homes the realtor will sell in a week?c.
Using the realtor sales information from Exercise 13.14, the realtor makes a typical commission of $4000 on each home sold, but he needs to pay an assistant $900 each week. a. What is the realtor's expected net profit each week? b. What is the standard deviation in the realtor's weekly net
Erika, a basketball player, scores an average of 10 points per game with a standard deviation of 2.6 points. Let X denote the number of points that she scores in a game. a. What is the expected value of X2? b. What is the variance of 24X - 3? c. What is the expected value and standard deviation of
A person makes an average of 3.12 phone calls a day with a standard deviation of 1.77 phone calls. Let X denote the number of calls made per day. a. What is the expected value of X2? b. What is the expected value of - 12X + 10? c. Find the variance of - 12X + 10.
Using the horse race information from Exercise 13.1. if Rosie's owner pays $500 for entering but wins $1000 for first place. $750 for second place, $500 for third place, and nothing for placing any lower.a. What are the expected winnings (or losses) for Rosie (i.e., including the entrance fee)?b.
An insurance company models the amount N of loss (in thousands) if a homeowner has at least one small claim. There is a constant k such that the mass of N is P(N = n) = k / 2n + 1 for n = 1,2,3,4 with no other possible choices for N. Find the value of k to make this a probability mass function.
The probability mass function you found in Exercise 13.20 was only for homeowners who had at least one small claim. The insurance company believes the probability that a homeowner will haw at least one small claim is 0.08. What is the probability mass function for amount of loss for all of their
A tour group is planning to pick a town to stay in when they get tired of driving. Since they don't know exactly where they will be staying, they don't know exactly how much the hotel rooms will cost. They will need 12 rooms. They know that individual hotel rooms along their route have an average
An airline intentionally overbooks its flight because it knows that that probability an individual passenger will not show up is 0.05 (assume passengers are independent). The plane can hold 80 people, and the airline sold 81 tickets for the flight. a. What is the probability that there will not be
A probability mass function for the class sizes at a small college is given in the table below (assume that these are the only allowed class sizes):Class sizeProbability10 ............ 0.0520 ............ 0.1530 ............ 0.4540 ............ 0.1050 ............ 0.1060 ............ 0.1070
A password is to be created at random by selecting 4 characters (with replacement) from the set {A, B,..., Z,0,1, .J.., 9}. Let X be the number of letters which are chosen for the password.a. Find the probability mass function for the number of letters which are chosen.b. Graph the probability mass
Assume that each password is created I according to Exercise 13.3. If passwords are independent and 10 passwords are I created:a. What is the expected total number of letters which are used?b. What is the standard deviation of the total number of letters which are used?c. What is the probability
There are 20 students competing to be on the Math Team. Thirteen of them specialize in algebra, 4 of them specialize in geometry, and 3 of them specialize in calculus. Only 5 spots on the team are available, and team members will be chosen at random, regardless of specialty. Let X be the number of
Joan will grade four statistics projects, selected at random from a large stack, this evening. From past experience, Joan thinks that 30% of the projects will be "A" quality projects. Let X be the number of "A" projects she grades tonight. (Assume that all the projects are independent and that Joan
For the following cumulative distribution function for a discrete random variable X,a. Find the probability mass function. Show why it is a valid probability mass function.b. What is the probability that X will be more than 2?c. Given that X is at most 2, what is the probability that it is
Which of the following statements are true about probability mass functions? Explain why each statement is true or false. a. P(X = x) can be 1 for a particular value of x. b. The graph of P(X = x) must be non-decreasing. c. P(X = x) can be negative for a particular value of x. d. P(X = x) can be 1
Which of the following statements are true about cumulative distribution functions? Explain why each statement is true or false.a. FX(x) can be 1 for a particular value of x.b. The graph of FX(x) must be non-decreasing.c. FX(x) can be negative for a particular value of x.d. FX(x) can be 1 for more
One out of every eight calls to your housel is from a family member. You will record whether the next call is from a family member. a. What do you consider a "success" in this story? What is its probability? b. What do you consider a "failure" in this story? What is its probability?! c. Why is this
Suppose that a person wins a game of chance with probability 0.40, and loses otherwise. If he wins, he earns 5 dollars, and if he loses, then he loses 4 dollars. a. What is his expected gain or loss? b. What is the variance of his gain or loss? c. Find constants a, b such that if X = 0 when he
You roll a fair, six-sided die as part of a game. If you roll a 5, you will win the game.a. What do you consider a "success" in this story? What is its probability?b. What do you consider a "failure" in this story? What is its probability?c. Why is this a Bernoulli situation? What is the
Hui has a class of 300 students, and only 6 have (lone their homework assignment due today. He calls on a student at random to put a problem on the board to check whether he or she has done the assignment.a. What does Hui consider a "success" in this story? What is the probability?b. What does Hui
Suppose that 1% of Blu-ray discs produced by a company are defective. You buy one of these discs and check to see if it is defective.a. What do you consider a "success" in this story? What is the probability?b. What do you consider a "failure" in this story? What is the probability?c. Why is this a
Anne and Jane have shoes spread throughout the dorm room. Anne has 15 pairs of shoes; twenty percent of her shoe collection consists of sandals. Jane has 40 pairs of shoes; ten percent of her shoe collection consists of sandals.a. A shoe is picked at random from the dorm room belonging to Anne and
Chris and Juanita always go to the movies on Friday night. Before meeting for their date, they each make a decision (independently, without consulting) of what genre of movie they prefer to see. Chris prefers an adventure movie with probability 70% and a romance with probability 30%; Juanita
Let X be the number of nights that you spend studying in a 30-day month. Assume that you study, on a given night, with probability 0.65, independent of the other nights. Write X as the sum of thirty indicators (i.e., as the sum of 30 Bernoulli random variables).a. Find E(X).b. Find Var(X).
As in Exercises 3.2, 10.2, 11.2, and 12.3, at a certain local restaurant, students are known to prefer Japanese pan noodles 40% of the time (it is a very popular and tasty dish!). a. Let X be an indicator for whether a randomly selected student orders Japanese pan noodles. Find E(X) and Var(X). b.
Skittles candies come in the colors red, orange, yellow, green, and purple, with each color having equal probability. You are quality control inspector, and your job is to count up the number of purple candies in a random sample of 25 candies from a large population of candies coming down the
Jeff typically makes 80% of his field goals. Steve typically makes 60% of his field goals. Suppose they both have the opportunity to kick 3 field goals. a. What is the probability Jeff will succeed in making at least 1 field goal? b. What is the probability Steve will succeed in making at least 1
Suppose a tennis player hits an ace once out of every live serves. Suppose in a match the player performs 80 serves. a. What is the expected number of aces? b. What are the variance and standard deviation of the number of aces?
On a multiple choice exam, a student decides to test his luck. His exam has 20 questions, each of which has 5 answer choices. The student decides to roll a die on each question and use the result on the die as his answer; any time that he rolls a 6, he just discards that roll and tries again. Let X
Two students decide to make bets about their plans for lunch during the next work week (Monday through Friday). They roll a six-sided die five times (once for each day). The agreement is that, for each day, when a 6 shows up, the first student has to pay for both lunches, and if a 1 shows up, the
Suppose that a person wins a game of chance with probability 0.40, and loses otherwise. If he wins, he earns 5 dollars, and if he loses, then he loses 4 dollars. Assume that he plays ten games independently. Let X denote the number of games that he wins. (His gain or loss is 5X + (-4)(10 - X) = 9X
Assume that when your phone rings, the caller is a telemarketer with probability 1/8, and that the probability of a telemarketer is independent from call to call. Let X denote the number of telemarketers during the next three calls.a. What is the mass of X?b. Draw a graph of the mass of X.c. Draw a
You randomly text 20 of your friends to see who wants to go on a hiking trip. You think that they all respond to your requests independently of each other, and you estimate that each one is 7% likely to be available, interested in going hiking, and will actually text you back to accept the
Let X, Y, Z be (respectively) the number of nights that Alice, Bob, and Charlotte eat in the dining hall during a 7-day week. Assume that X, Y, Z are independent Binomial random variables that each have n = 7 and p = 0.65. a. What is the distribution of X + Y + Z, i.e., the total number of meals
You draw seven cards, without replacement, from a muffled, standard deck of 52 playing cards. Let X be the number of hearts that are selected.a. What is the expected number of hearts? Why?b. Is X a Binomial random variable? Why or why not?
Approximately 8.33% of men are colorblind. You survey 8 men from the population of a large city and count the numbers who are colorblind.a. What is a "success" in this situation? What is the probability of a success on a single trial?b. What is a "failure" in this situation? What is the probability
There is a big exam tonight, and all of the 400 students are invited to attend the help session. From past experience, the instructor finds that 60% of the students are likely to attend the help session. She wants to reserve an appropriately sized room. a. What is the expected value and variance
Despite lecturing your roommates on energy conservation, there is a 60% chance that the lights in a dorm room will be left on when nobody is home. Each day is independent. Suppose that, every day the light is left on in a dorm room, there are 1000 Watts of power used. Every day when the light is
Samuel is an encyclopedia salesman. In order to make his life more interesting, when he encounters an intersection of two streets, he heads east 30% of the time and north the other 70%. He will walk 50 blocks (intersection-to-intersection) each day. West-East blocks contain 4 houses each and
In the ant world, 98% of the ants are female, and 2% are male. In the queen's first batch of 100,000 offspring, let X be the total number of male births. Write an expression for the probability that 2100 or more of the ants are males. You do not need to simplify or evaluate your expression. Would
Sixty percent of students usually eat cereal for breakfast. If n students eat in the dining halls for breakfast each day, let Xn be the number who have cereal. Find the limiting probability that at least one student has cereal, as n grows large, i.e., find limn→∞ P(Xn > 0). Does your answer
Twenty percent of babies in a particular city are born by a surgical procedure called a Cesarean section (C-section). You randomly survey 9 parents of babies from the population of the large city and count the number of babies who are not born by C-section.a. Explain in words what X is in this
A cereal company puts a Star Wars toy watch in each of its boxes as a sales promotion. Twenty percent of the cereal boxes contain a watch with Obi Wan Kenobi on it. You are a huge Obi Wan fan, so you really want one of these watches.a. You decide to buy 100 boxes of cereal from the warehouse store
A recent survey states that 48% of mobile devices are iPhones. In order to learn more about how the iPhone works, a student starts asking random people on campus if they use an iPhone. Assume that the individuals on campus are independent.a. If he surveys 20 people, what is the probability he finds
There are 7 books needing reshelving in the undergraduate library. Sixty-five percent of the library's collection consists of reference books. Let X be the number of reference books a student page reshelves out of the 7 on her cart.a. What is the probability that all 7 of them are reference
You have a 4-sidcd die with colors red, blue, yellow, and green for each face. You will roll the die 10 times. Let X be the number of rolls on which blue is the color selected.a. What is the probability blue is selected exactly 3 times?b. What is the probability blue is selected at most 3 times?c.
Skittles candies come in the colors red, orange, yellow, green, and purple, with each color having equal probability. Due to a dye mix-up there are a few rainbow-striped candies coming down the line. There is a 5% chance that a candy is rainbow striped. You are a quality control inspector, and your
Michael reaches into a very large box and pulls out Lucky Charms. If percentages of the pieces are: 50% regular cereal, 6.25% each for hearts, stars, horseshoes, clovers, blue moons, pots of gold, rainbows, and red balloons, and he only wants blue moons. Let X be the number of individual pieces he
Edward and his friend cannot go to school when it's sunny outside because they are vampires. Forks, Washington is experiencing a sunny period these days, and there is an 80% chance each day that it will be sunny. Assume that the weather is independent from day to day. Let .V be the number of days
A basketball player shoots free throws until he makes one. However, because his coach wants him to practice his technique and the feel of the motions, his coach has the player blindfold himself, which makes each throw independent from the others. When he is blindfolded, he has only a 2% chance of
Suppose that a person wins a game of chance with probability 0.40, and loses otherwise. He plays the game until he wins for the first time, and then he stops. Assume that the games are independent of each other. Let X denote the number of games that he must play until (and including) his first
Continue to use the scenario from the previous problem. As before, let X denote the number of games that he must play until (and including) his first win. Assume that, if he wins, he earns 5 dollars, and if he loses, then he loses 4 dollars. (Also assume that he is allowed to borrow money, i.e.,
You randomly call friends who could be potential partners for a dance. You think that they all respond to your requests independently of each other, and you estimate that each one is 7% likely to accept your request. Let X denote the number of phone calls that you make to successfully get a
Assume that when your phone rings, the caller is a telemarketer with probability 1/8, and that the probability of a telemarketer is independent from call to call. Let X denote the number of telemarketers during the next three calls. If n is a nonnegative integer, what is P(X > n)?
Prince Charming has to go around town asking if the glass slipper fits until he finds a woman whose foot fits properly in the slipper so that he will know who to marry. (We do not endorse this technique for finding a wife.) The probability of a glass slipper fitting a randomly selected woman is
Why does the memoryless property work for the Geometric distribution but not for the Binomial distribution?
Approximately 8.33% of men are colorblind. You survey men from a large population until you find one who is colorblind.a. Explain in words what X is in this situation and what values it can take.b. Why is this a Geometric distribution situation? What is the parameter?c. What is the probability you
Use the probability mass function to justify the fact that, if X is a Geometric random variable, then P(X > x) = qx. (In Section 16.2, we justified this intuitively, but we did not use the PMF to prove it.)
Twenty percent of babies in a particular city are born by a surgical procedure called a Cesarean section (C-section). You are interested in gathering information about hospital experiences of parents who did not have their baby by C-section. You survey randomly parents of babies from the population
(Refer to Exercise 15.6) A recent I survey states that 48% of mobile devices are iPhones. In order to learn more lout how the iPhone works, a student starts asking his friends if they use an I'liunc. Let X be the number of friends he will need to ask until he finds one 10 uses an iPhone.a. What is
A certain radio station plays songs from the 70's, 80's, and 90's. We know that 20% of the songs on the station are from the 70's; 37% are from the 80's; and 43% of the songs are from the 90's. Let be the number of songs you listen to until you hear the first one from the 90's. a. What is the
A young woman realizes that Homecoming is quickly approaching, and she needs to find a date. She estimates that 72% of the male students (in a large population) would be willing to accept her invitation. She plans to start randomly asking men for a date until someone accepts. a. What is the
Sarah and Thomas play a card game to determine who will have to take out the trash (the loser gets this unpopular chore) on Monday. They use a standard 52-card deck by taking turns randomly drawing cards from a shuffled deck, with replacement, until somebody draws an ace. Whoever gets the ace does
On a certain online math assignment, a student is allowed to submit an unlimited number of answers before moving on to the next problem. The problem is a free response question, and the student believes that his guessed answer is correct about 7% of the time. (There are so many possible guesses
Studies have shown that 28% of people do all of their Christmas shopping before Thanksgiving each year. Let X be the number of people you have to ask until you find somebody who has finished all of their Christmas shopping before Thanksgiving, assuming each person is independent of the others. a.
Skittles candies come in the colors red, orange, yellow, green, and purple, with each of these colors having equal probability. Due to a dye mix-up there are a few rainbow-striped candies coming down the line. There is a 5% chance that a candy is rainbow striped. You are a quality control
My alarm clock wakes me up only 64% of the time. My probability class meets at 8:30 am, and if I don't hear my alarm, I'll miss class. My teacher takes off one percent of our grade for every class that we miss. Let X be the number of class days that pass until I have lost 10% of my grade.a. What is
During a zombie apocalypse, one human finds that about 1 out of every 3 shots he makes actually kills a zombie. Let X be the number of shots he has to take until he kills his fifth zombie. a. Given that it takes between 14 and 16 shots (inclusive) to kill his fifth zombie, what is the probability
Philip and Callum are playing Monopoly. Callum has an 80% chance of winning whenever he plays Philip. If they play Monopoly until somebody wins 3 games (assuming no ties and that the games are independent): a. What is the probability that Callum wins the series in exactly 3 games? In 4 games? In 5
Approximately 8.33% of men are colorblind. You randomly survey men from a large population until you find 2 who are colorblind.a. Explain in words what X is in this situation and what values it can take.b. Why is this a Negative Binomial distribution situation? What are the parameters?c. What is
Twenty percent of babies in a particular city are born by a surgical procedure called a Cesarean section (C-section). You are interested in gathering information about hospital experiences of parents who did not have their baby by C-section. You randomly survey parents of babies from the population
A public relations intern realizes that she forgot to assemble the consumer panel her boss asked her to do. She panics and I decides to randomly ask (independent) people if they will work on the panel for an hour. Since she is willing to pay them for their work, she believes she willI have a 75%
Harry is a surfer who can successfully ride about 170% of waves. Assume that his wave-riding ability is independent, from wave to wave. Let X be the number of waves that pass until he successfully catches his 8th wave. What is the probability that X is between 10 and 12 (inclusive)?
You throw darts at a board with 20 equally likely spaces. You throw until you hit "1," then you throw until you hit "2," then you throw until you hit "3," etc., and finally you throw until you hit "20." Let X be the number of throws you make to achieve this goal.a. What are the expected value and
In the very large lecture class CHEM [115, 85% of students passed the first exam. I want to start a study group for students who failed the first test, with 10 students in the group. Because of FERPA privacy laws, I'm not allowed to get this information from the instructor. In surveying CHEM 115
Twelve percent of single women in the kingdom have feet which will fit into a glass slipper. Prince Charming thinks he must continue finding women who fit such a slipper, so that he has a collection to choose from. He would like 10 women who fit the slipper to compete on a "Bachelor"-type show for
You vow to replay a tough level in Super Mario World until you win 3 times. Assume that you have a 25% chance of winning each time you play, and each round is independent (your skill does not improve from game to game, because there is a lot of luck involved). Let X denote the number of times you
According to the Guinness Book of World Records (2005), there are 280 marriage licenses issued in Las Vegas per day. Let be the number of marriage licenses issued in the next 5 minutes.a. What values can X take? Why is this a Poisson situation? What is the parameter?b. What is the exact probability
Pedestrians walk by your store front frequently, but they only come into your store about once every 10 minutes on average. Let X be the number of pedestrians who enter the store in the next hour. a. Show the labeled graph of the mass for X. b. Show the labeled graph of the CDF for X.
Beetles walk across a lunch counter at an average rate of 5 beetles per hour.a. What is the probability that more than 2 beetles will cross the counter in the next half hour?b. What is the probability that more than 2 beetles will cross the counter in each of the next two half hours?c. What is the
People are competing to win a new car. The task, which only 1 in 5000 people can achieve, is to hit a golf ball into a cup 250 feet away. One hundred thousand people sign up to compete.a. How many people will win, on average? What is the standard deviation of the number of people who will win?b.
Customers arrive at a bakery at an average rate of f 6 per half-hour. For simplicity, suppose each customer spends $2.50 on their purchase.a. What is the probability that exactly 3 customers will arrive in the next 10 minutes?b. What is the probability that at least 3 customers will arrive in the
Workers at a factory produce a toy with a defect about once every 4 hours on average. Each toy costs the factory approximately $7 in labor and supplies. a. What is the expected number of toys with defects at the end of a 40-hour work week? b. What is the standard deviation in the number of toys
Workers at a factory produces toy with a defect about once every 4 hours on average. a. What is the probability there will be exactly 5 defects in 24 hours? b. What is the probability that there will be exactly 5 defects in each of the next seven 24-hour periods? c. What is the probability there
There are approximately 11,000 fish in a lake. Each fish has a 1 in 5500 chance of being albino. Let X be the number of albino fish.a. What is the expected number of albino fish in the lake?b. What is the approximate probability that there will be exactly 4 albino fish in the lake?c. Given that
A telemarketing firm uses a computer connected to an online phone book for Metropolis. A large number of messages are sent, but on average only 10 people per hour listen to such messages. Let X be the number of people who listen to these messages during a given hour. a. What is the probability that

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